Segre classes and Damon–Kempf–Laksov formula in algebraic cobordism

Abstract

In this paper, we introduce (relative) Segre classes for algebraic cobordism and prove a formula for their generating function. As an application, we prove a generalisation of the determinantal formula for the fundamental class of degeneracy loci to the algebraic cobordism of Grassmann bundles.

Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism

“…The goal of this section is to collect some basic properties of Borel-Moore homology theories and to translate in this context some of the results on generalised Segre classes appeared in [7].…”

“…Since in a general oriented cohomology theory not all Schubert varieties have a well defined notion of fundamental class, we opted for ψ * [Y C λ ] A as a replacement and, by combining Kazarian's approach with the generalised relative Segre classes S A introduced in [7], we obtained the following description.…”

“…The organisation of the paper is as follows. In section 2 we recall some basic facts about Borel-Moore homology theories, the covariant analogue of oriented cohomology theories, and we translate into this setting the results on Segre classes presented in [7]. In section 3 we prove the main theorem for symplectic Grassmann bundles, while in section 4 we first deal with the special case of quadric bundles, which we then use to establish the main theorem for odd orthogonal Grassmann bundles.…”

Identities on factorial Grothendieck polynomials

“…The factorial Grothendieck polynomial G λ (x|y) is the double Grothendieck polynomial corresponding to a Grassmannian permutation. These polynomials can also be interpreted in terms of set-valued tableaux [3,15,16,25,26], or expressed as the quotient of determinants [12][13][14]24]. The lowest degree homogeneous component of G λ (x|y) is equal to the factorial Schur function s λ (x|y), which is the double Schubert polynomial of a Grassmannian permutation and has received extensive attention, see, for example, [1,2,5,10,17,23,27,28].…”

“…On the other hand, there have been determinantal formulas for factorial Grothendieck polynomials [12][13][14]24]. For the purpose of this paper, we need the following determinantal formula for G λ (x|y) due to Ikeda and Naruse [14]:…”